# Difference between revisions of "ApCoCoA-1:Heisenberg groups"

### Heisenberg groups

#### Description

The Heisenberg group is the group of 3x3 upper triangular matrices of the form Heisenberg groups are often used in quantum mechanics and also occurs in fourier analysis. A representation is given by:

``` H(2k+1) = <a_{1},...,a_{k},b_{1},...,b_{k},c | [a_{i},b_{i}] = c, [a_{i},c] = [b_{i},c], [a_{i},b_{j}] = 1 for all i != j
```

#### Reference

Ernst Binz and Sonja Pods, Geometry of Heisenberg Groups, American Mathematical Society, 2008.

#### Computation

``` /*Use the ApCoCoA package ncpoly.*/

// Number of Heisenberg group
MEMORY.N:=1;

// a invers to d and b invers to e and c invers to f
Use ZZ/(2)[a[1..MEMORY.N],b[1..MEMORY.N],c,d[1..MEMORY.N],e[1..MEMORY.N],f];
NC.SetOrdering("LLEX");
Define CreateRelationsHeisenberg()
Relations:=[];
// add the relations of the invers elements ad = da = be = eb = cf = fc = 1
Append(Relations,[[c,f],]);
Append(Relations,[[f,c],]);
For Index1 := 1 To MEMORY.N Do
Append(Relations,[[a[Index1],d[Index1]],]);
Append(Relations,[[d[Index1],a[Index1]],]);
Append(Relations,[[b[Index1],e[Index1]],]);
Append(Relations,[[e[Index1],b[Index1]],]);
EndFor;

// add the relation [a_{i}, b_{i}] = c
For Index2 := 1 To MEMORY.N Do
Append(Relations,[[a[Index2],b[Index2],d[Index2],e[Index2]],[c]]);
EndFor;

// add the relation [a_{i}, c] = [b_i, c]
For Index3 := 1 To MEMORY.N Do
Append(Relations,[[a[Index3],c,d[Index3],f],[b[Index3],c,e[Index3],f]]);
EndFor;

// add the relation [a_{i}, b_{j}] = 1 for all i != j
For Index4 := 1 To MEMORY.N Do
For Index5 := 1 To MEMORY.N Do
If Index4 <> Index5 Then
Append(Relations,[[a[Index4],b[Index5],d[Index4],e[Index5]],]);
EndIf;
Endfor;
EndFor;

Return Relations;
EndDefine;

Relations:=CreateRelationsHeisenberg();
Relations;
Size(Relations);
GB:=NC.GB(Relations,31,1,100,1000);
Size(GB);
```